Rubinstein Bargaining

This lecture covers Rubinstein bargaining. In Rubinstein bargaining, the players take turns making offers over potentially infinitely many periods. Play continues until one player accepts the other’s offer. Every rejection leads to delay; we use discount factors to account for this.

Rubinstein bargaining is a desirable model to analyze because it feels realistic. In the previous games we have analyzed, negotiations arbitrarily terminated after one, two, or a few periods. That is not the case here. Furthermore, in real life, everyone has the ability to propose a deal. This model reflects that by allowing the person who rejects an offer to immediately propose a new one.

The game has a “no delay” solution—despite the possibility that the parties could negotiate forever, they reach a deal immediately. Intuitively, this is because any delay is costly. Thus, if we were to reach some division tomorrow, we could implement that same deal today and both do better because the immediate agreement does not discount the value of the bargain. The immediate agreement does not mean that the future periods are irrelevant. Rather, the types of agreements that would happen in the future determine the split of the benefits at the start.

The solution presented here leaves many questions unanswered. How does this game relate to the finite games we have studied earlier? How does the split change if the parties have different discount factors? Are there any other solutions? Future lectures will address these questions.

Note: Rubinstein gets its name from economist Ariel Rubinstein, who first found the solution to this game in a 1982 paper.